Some Diffraction Problems Involving Conical Geometries and their Rigorous Analysis

Abstract

The wave diffraction from the hollow finite and truncated perfectly conducting (rigid., soft) conical scatterers is considered. It is supposed that conical surfaces have zero thickness. The diffraction problems are formulated in the spherical coordinate system as the boundary value problems for the Helmholtz equation with respect to the scattered scalar potentials. The diffracted field is given by expansion in the series of eigenfunctions for subregions formed by the scatterers. Due to enforcement of the conditions of continuity together with the orthogonality properties of the Legendre functions the diffraction problems are reduced to infinite system of linear algebraic equations (i.s.l.a.e.). Usage of the analytical regularization approach transforms i.s.l.a.e. to the second kind and allows to justify the truncation method for obtaining numerical solution in the required class of sequences. These systems are proved to be regulated by a couple of operators, which consist of the convolution type operator and the corresponding inverted one. The elements of the inverted operator can be found analytically using the factorization technique.